具時滞的单种群模型和SIS模型的稳定性和分支分析
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第4章 SIS 传染病模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1 引言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 稳定性和 Hopf 分支的存在性. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Hopf 分支的方向和周期解的稳定性 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.4 数值模拟. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5 本章小结. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
第2章 预备知识 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 几何准则. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Hopf 分支存在条件 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
结 论. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 参考文献 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 攻读硕士学位期间发表的学术论文 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 哈尔滨工业大学硕士学位论文原创性声明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 哈尔滨工业大学硕士学位论文使用授权书 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 致 谢. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 个人简历 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
哈尔滨工业大学理学硕士学位论文
Abstract
Infectious disease dynamics is one important method for the theoretical quantitative investigation to the infectious disease. According to the characteristic of the population growth and disease occurrence, and the epidemic regularity in population, as well as its related social factors, the dynamics establishes qualitative, quantitative analysis and numerical simulation which can reflect that infectious disease dynamics characteristic to demonstrate the process of disease development, to promulgate its epidemic regularity, and forecast the potential of its change and development. It also could analyze the reason and the key aspect of the epidemic, seek the most superior control and prevention strategy, and finally provide the rationale and the quantitative basis for decision-making of the prevention and control.
第3章 成年种群增长模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 引言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 平衡点的稳定性 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 本章小结. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
The infectious disease model usually is considered about the existence and the stability of state of equilibrium (disease free of equilibrium and endemic disease state of equilibrium), the existence of periodic solution, the existence of bifurcation, establishing the basic reproduction number of disease infection. It is shown that the disease will not die out for the existence of periodic solution, it will change periodically; And the existence of bifurcation shows that the sensitivity of the disease spread, when some factors have small changes it is possible to cause radical changes for the condition of disease population. For example, the appearance of a Hopf bifurcation implies that the small change of a factor may cause huge changes of the situation to the disease. At last, The disease may break out periodically from a special state.
In this paper, a SIS epidemic model which is based on a population growth model is considered . The main contents of this paper are organized as follows. At first, we investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equations. Then formulas for determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are derived, using the normal from theory and center manifold argument. Finally, some numerical simulations are carried out for supporting the analytic found.
Keywords SIS epidemic model; Time-delay; Hopf bifurcation; Periodic solution
– –
哈尔滨工业大学理学硕士学位论文
目录
摘 要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
第1章 绪论 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 泛函微分方程 Hopf 分支理论的发展与研究现状 . . . . . . . . . . . . . . . . . . . . . 1 1.2 传染病模型的研究意义及国内外研究概况 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 传染病模型稳定性及分支的研究现状 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 本文的主要工作 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
第2章 预备知识 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 几何准则. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Hopf 分支存在条件 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
结 论. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 参考文献 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 攻读硕士学位期间发表的学术论文 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 哈尔滨工业大学硕士学位论文原创性声明 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 哈尔滨工业大学硕士学位论文使用授权书 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 致 谢. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 个人简历 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
哈尔滨工业大学理学硕士学位论文
Abstract
Infectious disease dynamics is one important method for the theoretical quantitative investigation to the infectious disease. According to the characteristic of the population growth and disease occurrence, and the epidemic regularity in population, as well as its related social factors, the dynamics establishes qualitative, quantitative analysis and numerical simulation which can reflect that infectious disease dynamics characteristic to demonstrate the process of disease development, to promulgate its epidemic regularity, and forecast the potential of its change and development. It also could analyze the reason and the key aspect of the epidemic, seek the most superior control and prevention strategy, and finally provide the rationale and the quantitative basis for decision-making of the prevention and control.
第3章 成年种群增长模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 引言 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 平衡点的稳定性 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 本章小结. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
The infectious disease model usually is considered about the existence and the stability of state of equilibrium (disease free of equilibrium and endemic disease state of equilibrium), the existence of periodic solution, the existence of bifurcation, establishing the basic reproduction number of disease infection. It is shown that the disease will not die out for the existence of periodic solution, it will change periodically; And the existence of bifurcation shows that the sensitivity of the disease spread, when some factors have small changes it is possible to cause radical changes for the condition of disease population. For example, the appearance of a Hopf bifurcation implies that the small change of a factor may cause huge changes of the situation to the disease. At last, The disease may break out periodically from a special state.
In this paper, a SIS epidemic model which is based on a population growth model is considered . The main contents of this paper are organized as follows. At first, we investigate the stability and Hopf bifurcation by analyzing the distribution of the roots of associated characteristic equations. Then formulas for determining the stability and the direction of periodic solutions bifurcating from Hopf bifurcations are derived, using the normal from theory and center manifold argument. Finally, some numerical simulations are carried out for supporting the analytic found.
Keywords SIS epidemic model; Time-delay; Hopf bifurcation; Periodic solution
– –
哈尔滨工业大学理学硕士学位论文
目录
摘 要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II
第1章 绪论 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 泛函微分方程 Hopf 分支理论的发展与研究现状 . . . . . . . . . . . . . . . . . . . . . 1 1.2 传染病模型的研究意义及国内外研究概况 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 传染病模型稳定性及分支的研究现状 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 本文的主要工作 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6